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Jules Hedges has written a thought-provoking blog post, On compositionality where he connects the familiar idea of compositionality to the idea of emergent effects in nature, where systems can be understood as either having compositional properties or emergent properties.

The key point about emergent systems is that they are hard to understand, and this is as true for engineering as it is for science. He goes on to say "As a final thought, I claim that compositionality is extremely delicate, and that it is so powerful that it is worth going to extreme lengths to achieve it", so that avoiding emergent effects is a characteristic of good programming language design.

Some thoughts:

His examples of emergent systems are biology and game theory from an economic perspective. I would add to this list physics: of his co-authored paper showing that the spectral gap is undecidable, David Pérez-García said "For example, our results show that adding even a single particle to a lump of matter, however large, could in principle dramatically change its properties."

Spolsky's famous characterisation of interfaces built on shaky foundations as Leaky abstractions to me makes the distinction between compositional and emergent systems a little less than sharp.

We could talk endlessly about the list of what he regards as compositionality-breaking features of PLs. The evils of global state are well-documented, but I find dmbarbour's argument that Local state is poison a very interesting alternative way to look at what properties do we want from code; more generally, what kind of compositionalty PLs offer is very much paradigm dependent. Gotos are considered harmful, but the Linux kernel has little trouble with longjmp because of its mandated coding style: compositionality in engineering is a not just a matter of semantics but also of use. He targets OO and Haskell's type classes - I think he is quite correct - note that within these paradigms one can regain compositionality by restricting to LSP or algebraic classes, and supporting his thesis we see that these ideas have spawned ideas for the design of new, cleaner PLs.